Groups, Geometry, and Dynamics

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Volume 11, Issue 1, 2017, pp. 245–289
DOI: 10.4171/GGD/396

Published online: 2017-04-20

Independence tuples and Deninger's problem

Ben Hayes[1]

(1) Vanderbilt University, Nashville, USA

We define modified versions of the independence tuples for sofic entropy developed in [22]. Our first modification uses an $\ell^{q}$-distance instead of an $\ell^{\infty}$-distance. It turns out this produces the same version of independence tuples (but for nontrivial reasons), and this allows one added flexibility. Our second modification considers the „action" a sofic approximation gives on $\{1,\dots,d_{i}\},$ and forces our independence sets $J_{i}\subseteq\{1,\dots,d_{i}\}$ to be such that $\chi_{J_{i}}-u_{d_{i}}(J_{i})$ (i.e. the projection of $\chi_{J_{i}}$ onto mean zero functions) spans a representation of $\Gamma$ weakly contained in the left regular representation. This modification is motivated by the results in [17]. Using both of these modified versions of independence tuples we prove that if $\Gamma$ is sofic, and $f\in M_{n}(\mathbb Z(\Gamma))\cap \mathrm {GL}_{n}(L(\Gamma))$ is not invertible in $M_{n}(\mathbb Z(\Gamma)),$ then det$_{L(\Gamma)}(f)>1.$ This extends a consequence of the work in [15] and [22] where one needed $f\in M_{n}(\mathbb Z(\Gamma))\cap \mathrm {GL}_{n}(\ell^{1}(\Gamma)).$ As a consequence of our work, we show that if $f\in M_{n}(\mathbb Z(\Gamma))\cap \mathrm {GL}_{n}(L(\Gamma))$ is not invertible in $M_{n}(\mathbb Z(\Gamma))$ then $\Gamma$ \actson $(\mathbb Z(\Gamma)^{\oplus n}/\mathbb Z(\Gamma)^{\oplus n}f)^{\widehat}$ has completely positive topological entropy with respect to any sofic approximation.

Keywords: Sofic groups, independence tuples, completely positive entropy, Fuglede–Kadison determinants

Hayes Ben: Independence tuples and Deninger's problem. Groups Geom. Dyn. 11 (2017), 245-289. doi: 10.4171/GGD/396